Continuous M-estimators and their interpolation by polynomials

Replacing the median by a general M-estimator, we construct in this paper a host of variants of the robust nonlinear pyramid transforms proposed by Donoho and Yu [SIAM J. Math. Anal., 31 ( 2000) pp. 1030-1061]. Some of these new variants are more amenable to numerical implementations with provable properties when compared to the Donoho-Yu median-based pyramid transforms. At the crux of this generalized construction is the following result: the inverse problem of interpolating a univariate polynomial of degree n with n + 1 prescribed values for any given continuous M-estimator on n + 1 nonoverlapping intervals is a well-posed procedure. While the proof of this result is nonconstructive, we study the use of Newton methods for constructing such a polynomial interpolant and report numerical results in some test cases.